Euclid's Elements
This page delivers no verdict; it presents claims and evidence with their sources. No theory in science is absolute. Method
For everyone
Euclid built geometry from a few basic assumptions (axioms and postulates), proving everything step by step. This "axiomatic method" served as the model for mathematics for more than two thousand years. The realisation that the fifth (parallel) postulate cannot be derived from the others gave rise to non-Euclidean geometries.
For specialists
The thirteen books of the Elements cover plane geometry, the theory of proportion (Eudoxus), number theory (infinitude of primes, IX.20; the Euclidean algorithm, VII.1–2) and solid geometry. The independence of the parallel postulate was shown by the hyperbolic geometry of Lobachevsky (1829) and Bolyai (1832), and rigorously by Beltrami's models. Hilbert (1899) gave a complete axiom system making Euclid's implicit assumptions explicit.
Claims
Evidence
Supporting
The proof of the infinitude of primes is in the Elements and is still accepted today.[1]
Consistent models of geometry have been built in which the negation of the parallel postulate holds (hyperbolic geometry).[2]
Challenging
Some of Euclid's proofs rely on unstated assumptions (e.g. that circles intersect, "betweenness"). These gaps were closed by Hilbert's axioms.[3]
Connections
Sources
- WebEuclid's Elements (online edition). Clark Universitymathcs.clarku.eduarchive
- EncyclopediaNineteenth Century Geometry. Stanford Encyclopedia of Philosophyplato.stanford.eduarchive
- Book (1899). Grundlagen der Geometrie. Teubner, Leipzig
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Cite this page
Athena contributors. (2026). Euclid's Elements. Athena: The tree of human knowledge. Retrieved November 1, 2026, from https://latifkedi.github.io/athena/en/d/oklid/
@misc{athena_oklid,
title = {Euclid's Elements},
author = {{Athena contributors}},
year = {2026},
howpublished = {\url{https://latifkedi.github.io/athena/en/d/oklid/}},
note = {Athena: The tree of human knowledge. Accessed: 2026-11-01}
}Added: September 27, 2026 · Last changed: September 27, 2026 · Page history